Newspace parameters
Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 273.bj (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.17991597518\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 11x^{14} + 88x^{12} - 303x^{10} + 758x^{8} - 968x^{6} + 867x^{4} - 30x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 11x^{14} + 88x^{12} - 303x^{10} + 758x^{8} - 968x^{6} + 867x^{4} - 30x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 84975 \nu^{14} - 847968 \nu^{12} + 6630625 \nu^{10} - 18977750 \nu^{8} + 45035320 \nu^{6} - 24480525 \nu^{4} + 847175 \nu^{2} + 141812432 ) / 47832247 \) |
\(\beta_{3}\) | \(=\) | \( ( 493812 \nu^{14} - 5094945 \nu^{12} + 38532300 \nu^{10} - 110284680 \nu^{8} + 203384572 \nu^{6} - 142262748 \nu^{4} + 4923156 \nu^{2} + \cdots - 24499860 ) / 47832247 \) |
\(\beta_{4}\) | \(=\) | \( ( 508068 \nu^{14} - 5088601 \nu^{12} + 39644700 \nu^{10} - 113468520 \nu^{8} + 257471965 \nu^{6} - 146369772 \nu^{4} + 5065284 \nu^{2} + \cdots + 227456047 ) / 47832247 \) |
\(\beta_{5}\) | \(=\) | \( ( - 516836 \nu^{14} + 6587845 \nu^{12} - 54823490 \nu^{10} + 227035283 \nu^{8} - 593353298 \nu^{6} + 862031072 \nu^{4} - 708141783 \nu^{2} + \cdots + 24504217 ) / 47832247 \) |
\(\beta_{6}\) | \(=\) | \( ( 593043 \nu^{15} - 5936569 \nu^{13} + 46275325 \nu^{11} - 132446270 \nu^{9} + 302507285 \nu^{7} - 170850297 \nu^{5} + 5912459 \nu^{3} + \cdots + 464932973 \nu ) / 47832247 \) |
\(\beta_{7}\) | \(=\) | \( ( 678018 \nu^{15} - 6784537 \nu^{13} + 52905950 \nu^{11} - 151424020 \nu^{9} + 347542605 \nu^{7} - 195330822 \nu^{5} + 6759634 \nu^{3} + \cdots + 702409899 \nu ) / 47832247 \) |
\(\beta_{8}\) | \(=\) | \( ( 1171830 \nu^{15} - 11879482 \nu^{13} + 91438250 \nu^{11} - 261708700 \nu^{9} + 550927177 \nu^{7} - 337593570 \nu^{5} + 11682790 \nu^{3} + \cdots + 677910039 \nu ) / 47832247 \) |
\(\beta_{9}\) | \(=\) | \( ( - 1684309 \nu^{15} + 18442424 \nu^{13} - 147371224 \nu^{11} + 503715002 \nu^{9} - 1257728472 \nu^{7} + 1585375792 \nu^{5} - 1435815378 \nu^{3} + \cdots + 1849848 \nu ) / 47832247 \) |
\(\beta_{10}\) | \(=\) | \( ( - 1684309 \nu^{14} + 18442424 \nu^{12} - 147371224 \nu^{10} + 503715002 \nu^{8} - 1257728472 \nu^{6} + 1585375792 \nu^{4} - 1435815378 \nu^{2} + \cdots + 49682095 ) / 47832247 \) |
\(\beta_{11}\) | \(=\) | \( ( - 4967952 \nu^{14} + 54479304 \nu^{12} - 435483047 \nu^{10} + 1492167256 \nu^{8} - 3728150096 \nu^{6} + 4731646851 \nu^{4} - 4258766712 \nu^{2} + \cdots + 147361976 ) / 47832247 \) |
\(\beta_{12}\) | \(=\) | \( ( - 7826720 \nu^{14} + 86276344 \nu^{12} - 690441745 \nu^{10} + 2385730760 \nu^{8} - 5973395120 \nu^{6} + 7707683038 \nu^{4} - 6838320936 \nu^{2} + \cdots + 236620312 ) / 47832247 \) |
\(\beta_{13}\) | \(=\) | \( ( 16756333 \nu^{15} - 183577065 \nu^{13} + 1466942565 \nu^{11} - 5017774290 \nu^{9} + 12519509445 \nu^{7} - 15780931770 \nu^{5} + \cdots - 18413505 \nu ) / 47832247 \) |
\(\beta_{14}\) | \(=\) | \( ( - 24499860 \nu^{15} + 269004648 \nu^{13} - 2150892735 \nu^{11} + 7384925280 \nu^{9} - 18460609200 \nu^{7} + 23512479908 \nu^{5} + \cdots + 730072644 \nu ) / 47832247 \) |
\(\beta_{15}\) | \(=\) | \( ( 24504217 \nu^{15} - 269029551 \nu^{13} + 2149783251 \nu^{11} - 7369954261 \nu^{9} + 18347161203 \nu^{7} - 23126728758 \nu^{5} + \cdots - 26984727 \nu ) / 47832247 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{11} - 3\beta_{10} - \beta_{2} + 3 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{14} + \beta_{13} - 5\beta_{9} - \beta_{7} \) |
\(\nu^{4}\) | \(=\) | \( -\beta_{12} + 7\beta_{11} - 16\beta_{10} \) |
\(\nu^{5}\) | \(=\) | \( 8\beta_{14} + 9\beta_{13} - 30\beta_{9} - 9\beta_{6} - 30\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( -9\beta_{4} + \beta_{3} + 48\beta_{2} - 99 \) |
\(\nu^{7}\) | \(=\) | \( \beta_{8} + 56\beta_{7} - 66\beta_{6} - 195\beta_1 \) |
\(\nu^{8}\) | \(=\) | \( 66\beta_{12} - 327\beta_{11} + 661\beta_{10} - 11\beta_{5} - 66\beta_{4} + 11\beta_{3} + 327\beta_{2} - 650 \) |
\(\nu^{9}\) | \(=\) | \( 11\beta_{15} - 382\beta_{14} - 459\beta_{13} + 1304\beta_{9} + 382\beta_{7} \) |
\(\nu^{10}\) | \(=\) | \( 459\beta_{12} - 2222\beta_{11} + 4448\beta_{10} - 88\beta_{5} \) |
\(\nu^{11}\) | \(=\) | \( 88\beta_{15} - 2593\beta_{14} - 3140\beta_{13} + 8804\beta_{9} - 88\beta_{8} + 3140\beta_{6} + 8804\beta_1 \) |
\(\nu^{12}\) | \(=\) | \( 3140\beta_{4} - 635\beta_{3} - 15084\beta_{2} + 29464 \) |
\(\nu^{13}\) | \(=\) | \( -635\beta_{8} - 17589\beta_{7} + 21364\beta_{6} + 59632\beta_1 \) |
\(\nu^{14}\) | \(=\) | \( - 21364 \beta_{12} + 102360 \beta_{11} - 204035 \beta_{10} + 4410 \beta_{5} + 21364 \beta_{4} - 4410 \beta_{3} - 102360 \beta_{2} + 199625 \) |
\(\nu^{15}\) | \(=\) | \( -4410\beta_{15} + 119314\beta_{14} + 145088\beta_{13} - 404345\beta_{9} - 119314\beta_{7} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).
\(n\) | \(92\) | \(106\) | \(157\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1 + \beta_{10}\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25.1 |
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−2.25575 | + | 1.30236i | −0.500000 | + | 0.866025i | 2.39228 | − | 4.14355i | 0.401974 | − | 0.232080i | − | 2.60472i | −2.25575 | − | 1.38260i | 7.25298i | −0.500000 | − | 0.866025i | −0.604503 | + | 1.04703i | |||||||||||||||||||||||||||||||||||||||||||||||||||
25.2 | −1.34967 | + | 0.779232i | −0.500000 | + | 0.866025i | 0.214404 | − | 0.371358i | 1.85159 | − | 1.06902i | − | 1.55846i | −1.34967 | − | 2.27561i | − | 2.44865i | −0.500000 | − | 0.866025i | −1.66602 | + | 2.88564i | |||||||||||||||||||||||||||||||||||||||||||||||||||
25.3 | −1.14630 | + | 0.661815i | −0.500000 | + | 0.866025i | −0.124000 | + | 0.214775i | −1.98394 | + | 1.14543i | − | 1.32363i | −1.14630 | + | 2.38453i | − | 2.97552i | −0.500000 | − | 0.866025i | 1.51612 | − | 2.62600i | |||||||||||||||||||||||||||||||||||||||||||||||||||
25.4 | −0.161178 | + | 0.0930563i | −0.500000 | + | 0.866025i | −0.982681 | + | 1.70205i | 2.28561 | − | 1.31960i | − | 0.186113i | −0.161178 | + | 2.64084i | − | 0.738004i | −0.500000 | − | 0.866025i | −0.245594 | + | 0.425381i | |||||||||||||||||||||||||||||||||||||||||||||||||||
25.5 | 0.161178 | − | 0.0930563i | −0.500000 | + | 0.866025i | −0.982681 | + | 1.70205i | −2.28561 | + | 1.31960i | 0.186113i | 0.161178 | − | 2.64084i | 0.738004i | −0.500000 | − | 0.866025i | −0.245594 | + | 0.425381i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
25.6 | 1.14630 | − | 0.661815i | −0.500000 | + | 0.866025i | −0.124000 | + | 0.214775i | 1.98394 | − | 1.14543i | 1.32363i | 1.14630 | − | 2.38453i | 2.97552i | −0.500000 | − | 0.866025i | 1.51612 | − | 2.62600i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
25.7 | 1.34967 | − | 0.779232i | −0.500000 | + | 0.866025i | 0.214404 | − | 0.371358i | −1.85159 | + | 1.06902i | 1.55846i | 1.34967 | + | 2.27561i | 2.44865i | −0.500000 | − | 0.866025i | −1.66602 | + | 2.88564i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
25.8 | 2.25575 | − | 1.30236i | −0.500000 | + | 0.866025i | 2.39228 | − | 4.14355i | −0.401974 | + | 0.232080i | 2.60472i | 2.25575 | + | 1.38260i | − | 7.25298i | −0.500000 | − | 0.866025i | −0.604503 | + | 1.04703i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
142.1 | −2.25575 | − | 1.30236i | −0.500000 | − | 0.866025i | 2.39228 | + | 4.14355i | 0.401974 | + | 0.232080i | 2.60472i | −2.25575 | + | 1.38260i | − | 7.25298i | −0.500000 | + | 0.866025i | −0.604503 | − | 1.04703i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
142.2 | −1.34967 | − | 0.779232i | −0.500000 | − | 0.866025i | 0.214404 | + | 0.371358i | 1.85159 | + | 1.06902i | 1.55846i | −1.34967 | + | 2.27561i | 2.44865i | −0.500000 | + | 0.866025i | −1.66602 | − | 2.88564i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
142.3 | −1.14630 | − | 0.661815i | −0.500000 | − | 0.866025i | −0.124000 | − | 0.214775i | −1.98394 | − | 1.14543i | 1.32363i | −1.14630 | − | 2.38453i | 2.97552i | −0.500000 | + | 0.866025i | 1.51612 | + | 2.62600i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
142.4 | −0.161178 | − | 0.0930563i | −0.500000 | − | 0.866025i | −0.982681 | − | 1.70205i | 2.28561 | + | 1.31960i | 0.186113i | −0.161178 | − | 2.64084i | 0.738004i | −0.500000 | + | 0.866025i | −0.245594 | − | 0.425381i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
142.5 | 0.161178 | + | 0.0930563i | −0.500000 | − | 0.866025i | −0.982681 | − | 1.70205i | −2.28561 | − | 1.31960i | − | 0.186113i | 0.161178 | + | 2.64084i | − | 0.738004i | −0.500000 | + | 0.866025i | −0.245594 | − | 0.425381i | |||||||||||||||||||||||||||||||||||||||||||||||||||
142.6 | 1.14630 | + | 0.661815i | −0.500000 | − | 0.866025i | −0.124000 | − | 0.214775i | 1.98394 | + | 1.14543i | − | 1.32363i | 1.14630 | + | 2.38453i | − | 2.97552i | −0.500000 | + | 0.866025i | 1.51612 | + | 2.62600i | |||||||||||||||||||||||||||||||||||||||||||||||||||
142.7 | 1.34967 | + | 0.779232i | −0.500000 | − | 0.866025i | 0.214404 | + | 0.371358i | −1.85159 | − | 1.06902i | − | 1.55846i | 1.34967 | − | 2.27561i | − | 2.44865i | −0.500000 | + | 0.866025i | −1.66602 | − | 2.88564i | |||||||||||||||||||||||||||||||||||||||||||||||||||
142.8 | 2.25575 | + | 1.30236i | −0.500000 | − | 0.866025i | 2.39228 | + | 4.14355i | −0.401974 | − | 0.232080i | − | 2.60472i | 2.25575 | − | 1.38260i | 7.25298i | −0.500000 | + | 0.866025i | −0.604503 | − | 1.04703i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
13.b | even | 2 | 1 | inner |
91.r | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 273.2.bj.c | ✓ | 16 |
3.b | odd | 2 | 1 | 819.2.dl.f | 16 | ||
7.c | even | 3 | 1 | inner | 273.2.bj.c | ✓ | 16 |
7.c | even | 3 | 1 | 1911.2.c.n | 8 | ||
7.d | odd | 6 | 1 | 1911.2.c.k | 8 | ||
13.b | even | 2 | 1 | inner | 273.2.bj.c | ✓ | 16 |
21.h | odd | 6 | 1 | 819.2.dl.f | 16 | ||
39.d | odd | 2 | 1 | 819.2.dl.f | 16 | ||
91.r | even | 6 | 1 | inner | 273.2.bj.c | ✓ | 16 |
91.r | even | 6 | 1 | 1911.2.c.n | 8 | ||
91.s | odd | 6 | 1 | 1911.2.c.k | 8 | ||
273.w | odd | 6 | 1 | 819.2.dl.f | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
273.2.bj.c | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
273.2.bj.c | ✓ | 16 | 7.c | even | 3 | 1 | inner |
273.2.bj.c | ✓ | 16 | 13.b | even | 2 | 1 | inner |
273.2.bj.c | ✓ | 16 | 91.r | even | 6 | 1 | inner |
819.2.dl.f | 16 | 3.b | odd | 2 | 1 | ||
819.2.dl.f | 16 | 21.h | odd | 6 | 1 | ||
819.2.dl.f | 16 | 39.d | odd | 2 | 1 | ||
819.2.dl.f | 16 | 273.w | odd | 6 | 1 | ||
1911.2.c.k | 8 | 7.d | odd | 6 | 1 | ||
1911.2.c.k | 8 | 91.s | odd | 6 | 1 | ||
1911.2.c.n | 8 | 7.c | even | 3 | 1 | ||
1911.2.c.n | 8 | 91.r | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 11T_{2}^{14} + 88T_{2}^{12} - 303T_{2}^{10} + 758T_{2}^{8} - 968T_{2}^{6} + 867T_{2}^{4} - 30T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} - 11 T^{14} + 88 T^{12} - 303 T^{10} + \cdots + 1 \)
$3$
\( (T^{2} + T + 1)^{8} \)
$5$
\( T^{16} - 17 T^{14} + 193 T^{12} + \cdots + 1296 \)
$7$
\( T^{16} + 23 T^{14} + 283 T^{12} + \cdots + 5764801 \)
$11$
\( T^{16} - 56 T^{14} + 2251 T^{12} + \cdots + 10556001 \)
$13$
\( (T^{8} - 2 T^{7} - 16 T^{6} - 14 T^{5} + \cdots + 28561)^{2} \)
$17$
\( (T^{8} + 4 T^{7} + 65 T^{6} + 294 T^{5} + \cdots + 28561)^{2} \)
$19$
\( T^{16} - 32 T^{14} + 682 T^{12} + \cdots + 923521 \)
$23$
\( (T^{8} + 4 T^{7} + 65 T^{6} - 82 T^{5} + \cdots + 244036)^{2} \)
$29$
\( (T^{4} + 9 T^{3} + 13 T^{2} - 32 T + 11)^{4} \)
$31$
\( T^{16} - 108 T^{14} + \cdots + 11316496 \)
$37$
\( T^{16} - 103 T^{14} + \cdots + 454371856 \)
$41$
\( (T^{8} + 100 T^{6} + 3089 T^{4} + \cdots + 7396)^{2} \)
$43$
\( (T^{4} - 8 T^{3} - 115 T^{2} + 1363 T - 3552)^{4} \)
$47$
\( T^{16} - 255 T^{14} + \cdots + 1442919878656 \)
$53$
\( (T^{8} - 18 T^{7} + 297 T^{6} + \cdots + 5517801)^{2} \)
$59$
\( T^{16} + \cdots + 909087685468561 \)
$61$
\( (T^{8} - 6 T^{7} + 101 T^{6} + \cdots + 1168561)^{2} \)
$67$
\( T^{16} - 180 T^{14} + \cdots + 1387488001 \)
$71$
\( (T^{8} + 332 T^{6} + 35851 T^{4} + \cdots + 4004001)^{2} \)
$73$
\( T^{16} - 284 T^{14} + \cdots + 2667616624656 \)
$79$
\( (T^{8} - 4 T^{7} + 301 T^{6} + \cdots + 199487376)^{2} \)
$83$
\( (T^{8} + 252 T^{6} + 16139 T^{4} + \cdots + 26244)^{2} \)
$89$
\( T^{16} - 53 T^{14} + 1947 T^{12} + \cdots + 1296 \)
$97$
\( (T^{8} + 63 T^{6} + 708 T^{4} + 2723 T^{2} + \cdots + 3364)^{2} \)
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